Question

The coordinates of $A,$ $B$ and $C$ are $(5,5),$ $(2,1)$ and $(0,k)$ respectively. The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible is:

$3$

$4\dfrac12$

$3\dfrac67$

$4\dfrac56$

$2\dfrac17$

Answer: E

Concepts:reflection shortest path coordinate geometry

Difficulty rating: 1910

Small Hint:

Reflect $B$ across the $y$ -axis so that $BC$ becomes the distance from $C$ to the reflected point

Big Hint:

The shortest broken path occurs where the straight line from $A$ to the reflected point meets the $y$ -axis

Solution:

Reflect $B=(2,1)$ across the $y$ -axis to $B'=(-2,1).$ For $C$ on the $y$ -axis, $BC=B'C,$ so $AC+BC$ is minimized when $A,C,B'$ are collinear. The line from $A=(5,5)$ to $B'=(-2,1)$ has slope $\frac{4}{7}.$ At $x=0,$ its height is $5-\frac47(5)=\frac{15}{7}=2\frac17.$ Hence $k=\frac{15}{7}.$

Thus, the correct answer is E.

1953 AMC 12 Problem 49

Answer: E

Difficulty rating: 1910

Small Hint:

Big Hint:

Solution:

Problem 49 in Other Years